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Main

  • Installation
  • Changelog
  • Breaking changes
  • Benchmarks
  • Acknowledgement
  • Bibliography

Tutorials

  • The basics
    • Taylor’s method
    • Introduction to the expression system
    • The adaptive integrator
    • Customising the adaptive integrator
    • ODEs with parameters
    • Non-autonomous systems
    • Dense & continuous output
  • Event detection
  • Variational ODEs
  • More on the expression system
    • Common pitfalls
    • Computing derivatives
    • Supporting large computational graphs
  • Parallelisation and vectorisation
    • Batch mode
    • Ensemble propagations
    • Parallel mode
    • Evaluating the performance of ensemble & batch mode
  • Extended and reduced precision
    • Computations in extended precision
    • Computations in arbitrary precision
    • Computations in single precision
  • Lagrangian and Hamiltonian mechanics
  • Interoperability with SymPy
  • Compiled functions
  • EOP and space weather data
    • Working with EOP data
    • Working with space weather data
  • Analytical theories and models
    • Introduction to the VSOP2013 planetary theory
    • Introduction to the ELP2000 lunar theory
    • Introduction to the IAU2000/2006 precession-nutation theory
    • Introduction to the EGM2008 model
  • Earth orbit dynamics
    • Third-body perturbations
    • Geopotential
    • Atmospheric drag
  • A differentiable SGP4 propagator
  • Pickle support
  • JIT compilation and caching

Examples

  • Celestial mechanics and astrodynamics
    • The three-body problem
      • The restricted three-body problem
      • Continuation of Periodic Orbits in the CR3BP
      • Pseudo arc-length continuation in the CR3BP
    • N-body and Solar System dynamics
      • Brouwer’s law in the outer Solar System
      • Long term stability of N-body simulations: the case of Trappist-1
      • Conserving first integrals via manifold projection
      • Planetary embryos in the inner Solar System
      • Mercury’s relativistic precession
      • Calculating transit timing variations
      • Elastic tides
    • Astrodynamics
      • Box control in satellite Formation Flying
      • Comparing coordinate systems
      • Inverting Kepler’s equation in ODEs
      • Lagrange propagation and the state transition matrix
      • Gravity-gradient stabilization
  • Event detection
    • Sampling events
    • Poincaré sections
    • The second integral of motion
    • The Keplerian billiard
    • The two-fixed centres elliptic billiard
    • The wavy ramp
    • The Maxwell-Boltzmann distribution
    • Computing event sensitivity
  • Machine Learning
    • Feed-Forward Neural Networks
    • Interfacing torch to heyoka.py
    • Neural ODEs - I
    • Neural ODEs - II
    • Neural Hamiltonian ODEs
    • ThermoNETs
    • Differentiable Atmosphere
  • Variational equations
    • Jet transport in the simple pendulum
    • Learning mass distributions in asteroids
    • Map inversion and event sensitivities
    • The Tolman–Oppenheimer–Volkoff equations (Lindblom’s form)
    • Computing collision probabilities with ETTs
  • Others
    • The variational equations
    • Optimal Control of the Lotka-Volterra equations
    • Computing definite integrals

API reference

  • Common keyword arguments
  • Expression system
    • heyoka.expression
    • heyoka.func_args
    • heyoka.dtens
    • heyoka.make_vars
    • heyoka.diff_tensors
    • heyoka.subs
    • heyoka.sum
    • heyoka.prod
    • heyoka.par
    • heyoka.time
    • heyoka.diff_args
  • Numerical integrators
    • heyoka.taylor_adaptive
    • heyoka.taylor_adaptive_batch
  • Variational ODE systems
    • heyoka.var_ode_sys
    • heyoka.var_args
  • Lagrangian and Hamiltonian mechanics
    • heyoka.lagrangian
    • heyoka.hamiltonian
  • Models
    • Dynamics
      • heyoka.model.fixed_centres
      • heyoka.model.pendulum
    • Coordinate transformations
      • heyoka.model.cart2geo
      • heyoka.model.geo2cart
      • heyoka.model.rot_fk5j2000_icrs
      • heyoka.model.rot_icrs_fk5j2000
      • heyoka.model.rot_itrs_icrs
      • heyoka.model.rot_icrs_itrs
      • heyoka.model.rot_itrs_teme
      • heyoka.model.rot_teme_itrs
      • heyoka.model.state_to_rsw
      • heyoka.model.state_to_rsw_inertial
      • heyoka.model.state_from_rsw
      • heyoka.model.state_from_rsw_inertial
    • Time transformations
      • heyoka.model.delta_tdb_tt
      • heyoka.model.dayfrac
      • heyoka.model.delta_tt_tai
    • Atmospheric models
      • heyoka.model.nrlmsise00_tn
      • heyoka.model.jb08_tn
    • Earth orientation and space weather
      • heyoka.model.era
      • heyoka.model.erap
      • heyoka.model.gmst82
      • heyoka.model.gmst82p
      • heyoka.model.pm_x
      • heyoka.model.pm_xp
      • heyoka.model.pm_y
      • heyoka.model.pm_yp
      • heyoka.model.dX
      • heyoka.model.dXp
      • heyoka.model.dY
      • heyoka.model.dYp
      • heyoka.model.Ap_avg
      • heyoka.model.f107
      • heyoka.model.f107a_center81
    • Analytical theories and models
      • heyoka.model.iau2006
      • heyoka.model.egm2008_pot
      • heyoka.model.egm2008_acc
      • heyoka.model.get_egm2008_mu
      • heyoka.model.get_egm2008_a
      • heyoka.model.vsop2013_cartesian_icrf
      • heyoka.model.vsop2013_cartesian
      • heyoka.model.vsop2013_elliptic
      • heyoka.model.get_vsop2013_mus
      • heyoka.model.elp2000_cartesian_fk5
      • heyoka.model.elp2000_cartesian_e2000
      • heyoka.model.get_elp2000_mus
    • SGP4 propagation
      • heyoka.model.sgp4_propagator_dbl
      • heyoka.model.sgp4_propagator_flt
      • heyoka.model.sgp4
      • heyoka.model.gpe_is_deep_space
      • heyoka.model.sgp4_propagator
  • EOP and space weather data
    • heyoka.eop_data
    • heyoka.sw_data
    • heyoka.eop_data_row
    • heyoka.sw_data_row
  • JIT compilation
    • heyoka.llvm_state
    • heyoka.llvm_multi_state
    • heyoka.code_model
  • Repository
  • Open issue
  • .rst

More on the expression system

More on the expression system#

  • Common pitfalls
  • Computing derivatives
  • Supporting large computational graphs

previous

Variational ODEs

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Common pitfalls

By Francesco Biscani and Dario Izzo

© Copyright 2020-2025 Francesco Biscani and Dario Izzo.